Lie algebra–valued differential form

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Formal definition

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A Lie-algebra-valued differential  -form on a manifold,  , is a smooth section of the bundle  , where   is a Lie algebra,   is the cotangent bundle of   and   denotes the   exterior power.

Wedge product

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The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers. For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form. For a  -valued  -form   and a  -valued  -form  , their wedge product   is given by

 

where the  's are tangent vectors. The notation is meant to indicate both operations involved. For example, if   and   are Lie-algebra-valued one forms, then one has

 

The operation   can also be defined as the bilinear operation on   satisfying

 

for all   and  .

Some authors have used the notation   instead of  . The notation  , which resembles a commutator, is justified by the fact that if the Lie algebra   is a matrix algebra then   is nothing but the graded commutator of   and  , i. e. if   and   then

 

where   are wedge products formed using the matrix multiplication on  .

Operations

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Let   be a Lie algebra homomorphism. If   is a  -valued form on a manifold, then   is an  -valued form on the same manifold obtained by applying   to the values of  :  .

Similarly, if   is a multilinear functional on  , then one puts[1]

 

where   and   are  -valued  -forms. Moreover, given a vector space  , the same formula can be used to define the  -valued form   when

 

is a multilinear map,   is a  -valued form and   is a  -valued form. Note that, when

 

giving   amounts to giving an action of   on  ; i.e.,   determines the representation

 

and, conversely, any representation   determines   with the condition  . For example, if   (the bracket of  ), then we recover the definition of   given above, with  , the adjoint representation. (Note the relation between   and   above is thus like the relation between a bracket and  .)

In general, if   is a  -valued  -form and   is a  -valued  -form, then one more commonly writes   when  . Explicitly,

 

With this notation, one has for example:

 .

Example: If   is a  -valued one-form (for example, a connection form),   a representation of   on a vector space   and   a  -valued zero-form, then

 [2]

Forms with values in an adjoint bundle

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Let   be a smooth principal bundle with structure group   and  .   acts on   via adjoint representation and so one can form the associated bundle:

 

Any  -valued forms on the base space of   are in a natural one-to-one correspondence with any tensorial forms on   of adjoint type.

See also

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Notes

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  1. ^ S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2. Chapter XII, § 1.}}
  2. ^ Since  , we have that
     
    is  

References

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